Page 132 - Mathematical Techniques of Fractional Order Systems

120 Mathematical Techniques of Fractional Order Systems

Taking the Laplace transform of (4.87) and using the time shift property
of the Laplace transform, we obtain

8 9
2λlnα
< ð t1lnαÞ =
1 22lnα

Ð
L t-s vtðÞ 5 e Ht 1 lnαÞ dα

ð
0 L t-s
Γ 2λlnα 1 1Þ
ð
: ;
1
1 22lnα
Ð
5 e e slnα dα
0 2λlnα11
s ð4:88Þ
5 Ð 1 22lnα 22λlnα11ð Þlns slnα dα
e
e
e
0
5 Ð 1 ð 221λlns1sÞlnα dα=s
e
0
Ð 1
5 α ð 221λlns1sÞ dα=s
0
The resultant expression under the condition Re s 2 2 1 λlnsg . 0 equals
f
1

L t-s vtðÞ 5 ð4:89Þ
ss 2 1 1 λlnsÞ
ð
On the other hand, it can be shown that solving (4.82) for XsðÞ in the
Laplace domain results
s 2 1
XsðÞ 5 x 0ðÞ ð4:90Þ
ð
ss 2 1 1 λlnsÞ
From (4.89) and (4.90), it is deduced that
0
xtðÞ 5 v tðÞ 2 vtðÞð Þx 0ðÞ ð4:91Þ
Calculating the derivative d vtðÞ from (4.85) results
dt
ð t ð t2τÞ λτ21
τ
v tðÞ 5 e dτ ð4:92Þ
0
ð
0 ΓλτÞ
in which t . 0. This Theorem is proved by replacing vtðÞ and v tðÞ in (4.91)
0
with their expressions derived in (4.85) and (4.92), respectively.
A similar result may be stated in case of exponential weight functions as
in the following corollary.
α
Corollary 2: Exact solution of (4.82) under the assumption that w αðÞ 5 ca ,
αA 0; 1Š, aAR . 0 , and cAR 2 0fg holds, is given by
½
λτ λτ !
ð t=a t=a2τ c 21 t=a2τ c
τ
xtðÞ 5 x 0ðÞ 2 e dτ ð4:93Þ
0 Γλτ=c Γλτ=c 1 1
Proof: Solution (4.93) is verified by following a procedure similar to the
one presented in Corollary 1.
We conclude this section by presenting a new integral identity that
directly follows from the results of this section as a side result. In fact